3.1.4 Sum/Product of the Roots

Define a polynomial by $p_n(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x^1 + a_0 = 0$. The three most popular questions associated with the number sense test concerning roots of polynomials are: sum of the roots, sum of the roots taken two at a time, and product of the roots. For the polynomial $p_n(x)$ these values are defined:

• Sum of the roots: $-\frac{a_{n-1}}{a_n}$
• Sum of the roots taken two at a time: $\frac{a_{n-2}}{a_n}$
• Product of the roots:
  • If $n$ is even: $\frac{a_0}{a_n}$
  • If $n$ is odd: $-\frac{a_0}{a_n}$

Let’s see what this means for our generic quadratics/cubics:

$p_2(x) = ax^2 + bx + c = 0$ and $p_3(x) = ax^3 + bx^2 + cx + d = 0$

$p_2(x) = ax^2 + bx + c = 0$

• Sum of the roots: $-\frac{b}{a}$
• Product of the roots: $\frac{c}{a}$

$p_3(x) = ax^3 + bx^2 + cx + d = 0$

• Sum of the roots: $-\frac{b}{a}$
• Product of the roots taken two at a time: $\frac{c}{a}$
• Product of the roots: $-\frac{d}{a}$

Since the quadratic only has two roots, the sum of the roots taken two at a time happens to be the product of the roots. You can extend the same procedure for polynomials of any degree, keeping in mind the alternating signs for the product of the roots. The following are practice problems:

Problem Set 3.1.4